To find [tex]\(\tan 2\theta\)[/tex] given that [tex]\(\sin \theta = \frac{8}{17}\)[/tex] and [tex]\(\theta\)[/tex] is in quadrant 2, let's go through the steps systematically.
1. Determine [tex]\(\sin \theta\)[/tex]:
We are given that [tex]\(\sin \theta = \frac{8}{17}\)[/tex].
2. Determine [tex]\(\cos \theta\)[/tex]:
Since [tex]\(\theta\)[/tex] is in quadrant 2, [tex]\(\cos \theta\)[/tex] is negative. We use the Pythagorean identity:
[tex]\[
\cos^2 \theta + \sin^2 \theta = 1
\][/tex]
Plugging in the value for [tex]\(\sin \theta\)[/tex]:
[tex]\[
\cos^2 \theta + \left( \frac{8}{17} \right)^2 = 1
\][/tex]
[tex]\[
\cos^2 \theta + \frac{64}{289} = 1
\][/tex]
[tex]\[
\cos^2 \theta = 1 - \frac{64}{289}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{289}{289} - \frac{64}{289}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{225}{289}
\][/tex]
[tex]\(\cos \theta\)[/tex] is the square root of [tex]\(\frac{225}{289}\)[/tex], but since [tex]\(\theta\)[/tex] is in quadrant 2, [tex]\(\cos \theta\)[/tex] is negative:
[tex]\[
\cos \theta = -\sqrt{\frac{225}{289}} = -\frac{15}{17}
\][/tex]
3. Determine [tex]\(\tan \theta\)[/tex]:
The tangent of [tex]\(\theta\)[/tex] is given by:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
Substituting the values:
[tex]\[
\tan \theta = \frac{\frac{8}{17}}{-\frac{15}{17}} = -\frac{8}{15}
\][/tex]
4. Use the double-angle formula for tangent:
The formula for [tex]\(\tan 2\theta\)[/tex] is:
[tex]\[
\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}
\][/tex]
Substituting [tex]\(\tan \theta = -\frac{8}{15}\)[/tex]:
[tex]\[
\tan 2\theta = \frac{2 \left( -\frac{8}{15} \right)}{1 - \left( -\frac{8}{15} \right)^2}
\][/tex]
[tex]\[
\tan 2\theta = \frac{-\frac{16}{15}}{1 - \frac{64}{225}}
\][/tex]
Simplifying the denominator:
[tex]\[
1 - \frac{64}{225} = \frac{225}{225} - \frac{64}{225} = \frac{161}{225}
\][/tex]
So,
[tex]\[
\tan 2\theta = \frac{-\frac{16}{15}}{\frac{161}{225}}
\][/tex]
Inverting and multiplying:
[tex]\[
\tan 2\theta = -\frac{16}{15} \cdot \frac{225}{161} = -\frac{16 \cdot 225}{15 \cdot 161} = -\frac{3600}{2415} = -\frac{24}{16} = -1.49 \text{ (rounded to 2 decimal places)}
\][/tex]
Thus, [tex]\(\tan 2\theta \approx -1.49\)[/tex].
